Lebesgue 测度

Preliminary

• L-S-measure

Definitions

Define $d$-ary function $F:{\mathbb{R}}^d\to \mathbb{R},F(x_1,x_2,\cdots ,x_d):=x_1{x}_2\cdots x_d$

It is easy to see that $F$ is an L-S function, which is called Lebesgue function, abbreviated as L function. The L-S measure induced by the $d$-dimensional L function on $\mathcal{B}({\mathbb{R}}^d)$ is called Lebesgue measure on $\mathcal{B}({\mathbb{R}}^d)$, abbreviated as $L$ measure, denoted as $\lambda$, thus $({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d),\lambda )$ is a measure space.

${\lambda }^{\ast }$ is the outer measure in measure extension from semi-ring to sigma algebra.

Let $({\mathbb{R}}^d,\overline{\mathcal{B}({\mathbb{R}}^d)},\overline{\lambda })$ be the completion of $({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d),\lambda )$. $\overline{\mathcal{B}({\mathbb{R}}^d)}=\sigma (\mathcal{B}({\mathbb{R}}^d)\cup {\mathcal{N}}_{\lambda })$ is called Lebesgue measurable set, abbreviated as L measurable set.

$A\subseteq {\mathbb{R}}^d,y\in {\mathbb{R}}^d,A+y:=\{a+y:a\in A\}$
$A\subseteq {\mathbb{R}}^d,-A:=\{-a:a\in A\}$

Properties

1. $x\in A⟺x+y\in A+y$
2. $x-y\notin A⟺x\notin A+y$
3. $x\in A⟺-x\in -A$
4. $x\notin A⟺-x\notin -A$
5. $A\cap (B+y)=(A-y)\cap B+y$
6. $A^c+y=(A+y{)}^c$
7. ${\lambda }^{\ast }(A+y)={\lambda }^{\ast }(A)$
8. ${\lambda }^{\ast }(A)={\lambda }^{\ast }(-A)$
9. $x\in -(A^c)⟺x\in (-A{)}^c$
10. L measure is invariant under translation, i.e. $\forall A\subseteq {\mathbb{R}}^d,x\in {\mathbb{R}}^d$, ($A$ is L measurable $⟺$ $A+y$ is L measurable), and $\lambda (A+y)=\lambda (A)$
11. L measure is invariant under reflection, i.e. $\forall A\subseteq {\mathbb{R}}^d$, ($A$ is L measurable $⟺$ $-A$ is L measurable)
12. $\exists A\subseteq \mathbb{R}$, s.t. $A$ is not L measurable. For example, Vitali set.